# A Theorem on Constructing Random Variables

I have in mind a theorem. I think it's straightforward but tedious to prove carefully and I'd like to use it to answer another question. Has anyone heard of this theorem? Does anyone know of a quick (non-tedious!) proof?

Suppose $(\Omega,\Sigma,\mathbb{P})$ is a probability space. Suppose $\mathcal{P}_1,\mathcal{P}_2,\ldots$ is a sequence of increasingly fine finite partitions of $\Omega$. That is, each $\mathcal{P}_n$ is a finite partition of $\Omega$ and if $A\in\mathcal{P}_{n+1}$, then $A\subset A'$ for some $A'\in\mathcal{P}_n$. And the big assumption: Suppose for each $\epsilon>0$, there is an $n$ such that $\mathbb{P}(A)<\epsilon$ for all $A\in\mathcal{P}_n$. Then for any possible cumulative distribution function $F:\mathbb{R}\to[0,1]$ (i.e., increasing, right continuous, $F(-\infty)=0$, $F(\infty)=1$), there is a random variable $X$ on $(\Omega,\Sigma,\mathbb{P})$ with $F$ as its cdf.

I am wondering if there's a middle ground between "it can be shown" and a proof by direct construction, which is tedious.

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